Abstract

The “plate-diagram” method of quantifying and manipulating the Seidel aberrations of an optical system has been used to develop a procedure that has successfully determined the complete solution set of three-mirror anastigmats in which two surfaces are left strictly spherical. The procedure also readily identified solutions in which the Petzval sum is zero, and four distinct families of flat-field three-mirror anastigmats with two mirrors strictly spherical have thus been found. The success of the method is strong support for the argument that algebraic approaches to optical design can yield results distinctly superior to currently favored optimization-based design methods, at least for some types of optical systems.

Ray reflected at a convex surface. All quantities shown here are positive. Note that P in this case is the length of the perpendicular from the center of curvature of the mirror to the incident ray. All other quantities follow the usual paraxial conventions as given by Conrady.10

Solution set for three-mirror anastigmats obtained by using Eq. (23) with the primary mirror aspherized. In this, and in the following figures representing solution sets, white points represent solutions with positive Petzval curvature, black points represent systems with negative Petzval curvature, and gray points represent coordinates for which no physically realizable anastigmats exist. Note that flat-field “Paul–Rumsey” systems can be found along a curve defined by where a white region abuts the black region in this graph.

Solution set for three-mirror anastigmats obtained by using Eq. (24) with the primary mirror aspherized. In this case the secondary mirror is concave (c2 is negative) for all solutions. No flat-field solutions exist in this set.

Solution set for three-mirror anastigmats obtained by using Eq. (23) with the secondary mirror aspherized, representing systems that have not previously been described in the literature. This graph corresponds closely to that shown in Fig. 1. Here there are two flat-field curves.

Solution set for three-mirror anastigmats obtained by using the method described in Section 4. Here the primary and secondary mirrors are spherical and the tertiary is aspherized. No flat-field solutions exist in this set.